Since von Neumann's axiomatization of quantum mechanics in the 1930s, measurement has been a kind of stepchild to unitary evolution. s the link between quantum mechanics and the classical world, measurement has attracted considerable skeptical scrutiny from philosophers. In the domain of quantum computation, also, measurement is often dreaded: an act which may project out degrees of computational freedom and potentially decoheres useful quantum correlations. However, from a pragmatic condensed matter perspective, measurement is a co-equal pillar of quantum mechanics and a tool to be exploited. In mathematics, also, unitaries and projectors are dual. Bott periodicity states that loops of unitaries correspond to a projector and a loop of projectors corresponds to a unitary. This perspective links the Chem class in the bulk to the energy current at the edge in the integral quantum Hall systems.
It is a familiar idea that measurement can stop something from happening, e.g., the “quantum watched pot effect” of Zeno's paradox. It is less familiar that measurement can elicit an intended evolution of states. To get an idea for how this might work, consider adiabatic evolution of a vector Ψ in a degenerate ground state manifold of a Hamiltonian H. Perturbing the Hamiltonian in time, H(t), while leaving the degeneracy k intact, will evolve Ψ in time according to the canonical connection on the “tautological bundle” over the Grassmann of k-planes. A discrete description of this evolution amounts to moving the k-plane slightly, leaving Ψ behind, and then projecting Ψ orthogonally back into the moved plane and repeating. It is well known that adiabatic evolution can affect the general unitary on the ground space k-plane, so a composition of projections (“measurements”) suffices to simulate unitary evolution in this simple example. The example amounts to the “quantum watched pot effect” where the “pot” is not holding still but evolving.
The preceding example suggests that a quantum state can be deliberately nudged along by a sequence of measurements as an alternative to (directly) constructing a unitary evolution of a state's underlying degrees of freedom. In quantum computation, the accuracy of the evolution is paramount. A principle advantage of the topological model is that the unitaries corresponding to braid representations are essentially exact (topologically protected). However, the topological model also anticipates measuring states in the basis of “topological charge,” and this basis is also topologically protected. Furthermore, the prime tool for such measurements, quasiparticle interferometry, is rapidly developing in both theory and experiment. An important result is that the operation of Fabry-Pérot interferometers in the Fractional Quantum Hall (FQH) context produce density matrices which converge exponentially fast to projection onto charge sectors (plus an additional projection “severing charge lines” running from the interior to exterior of the interferometer). With this tool in hand, it makes sense to ask whether we can organize universal quantum computation, again in the context of FQH fluids, as a sequence of interferometrical measurements, rather than as an exercise in braiding of quasiparticles, which has been the hypothetical paradigm since 2000. Actually, the older paradigm also required a bit of measurement, presumably interferometry, to properly initialize the system and then to measure its computational output. What we find is the topological model of computation can, in fact, be disencumbered from the necessity of braiding. From a technological point of view, this may be an important advance since much has been done experimentally with interferometers.
At some level, it should not be a surprise that interferometry can substitute for braiding. After all, an interferometer does braid a stream of quasiparticles running along the system edge (and tunneling across junctions) around other quasiparticles fixed in the bulk. At bottom, the procedures look similar. However, a little more thought turns up a conundrum: How can an operation which reduces rank (projection) simulate an operation which has full rank (unitary evolution)? The crucial answer is that we do not attempt to simulate the braid evolution of the general state vector Ψ through measurements. We only need to simulate the effect of the braid evolution on our initial state Ψ0. That is all we ever ask of a quantum computer: to faithfully evolve a known initial state.